Neft Teacher · Learn It

Unit 6 · Standard 6.AT.7

The Distributive Property

Lesson 6-14 · Learn It — how the math works

Why we're learning this

You're selling concert bundles at the music studio. Each bundle includes a ticket ($15) and a snack ($5). If 3 people each buy a bundle, you can calculate the total as 3(15 + 5) OR as 3 × 15 + 3 × 5. Either way, the total is the same — $60! This is the distributive property.

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Learn It

What is the distributive property?

The distributive property means you can multiply a number by everything inside the parentheses, one piece at a time: a(b + c) = ab + ac.

Picture itSee what the math looks like before you read about it

The studio manager is ordering merch bundles for 9 band members going on tour. Each bundle includes a t-shirt ($18) and a strap ($7). The order form writes the cost two ways: 9(18 + 7) and 9 × 18 + 9 × 7.
Where you see this in real lifeThe studio manager is ordering merch bundles for 9 band members going on tour. Each bundle includes a t-shirt ($18) and a strap ($7). The order form writes the cost two ways: 9(18 + 7) and 9 × 18 + 9 × 7.

Watch me solve itI do — read one step at a time

The problem looks like this
Tape diagram: 3 bundles, each a $15 ticket and a $5 snack, make $60 in all, so 3(15 + 5) = 3 × 15 + 3 × 5.6015ticket $155snack $5155155
  1. Each bundle has a ticket ($15) and a snack ($5). For 3 bundles I can write 3(15 + 5).
  2. Way 1: add inside first. 15 + 5 = 20, then 3 × 20 = 60.15 + 5 = 20
  3. Way 2: distribute the 3 to each part. 3 × 15 + 3 × 5.part. 3 × 15 + 3 × 5
  4. That is 45 + 15 = 60.45 + 15 = 60
  5. Both ways give $60, so 3(15 + 5) = 3 × 15 + 3 × 5.

Watch out — mistakes students make here

  • The 7 reached the x but skipped the 3. Distributing multiplies EVERY term inside — what is 7 × 3?
  • The 10 comes from ADDING 7 + 3. Distributing multiplies across, so the constant term should be 7 × 3.
  • The 7 reached the 3 but skipped the x. Both terms inside the parentheses get multiplied by 7.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Let's 2(8 + 3). Where does the 2 go?
  2. We multiply the 2 by each term:2 × 8 + 2 × 3
  3. That is 16 + 6 = 22. So 2(8 + 3) = 22.16 + 6 = 22

On your ownYou do — solve this one by yourself

Before you start, remember:

  • In the table, each expression using , then simplify.
  • Start with 2(8 + 3) = 2 × 8 + 2 × 3.

Solve: Use to each expression.

Factored FormExpanded Form
3(2x + 5)
7(a − 4)
2(3m + 8)
5(6 − y)

Stuck? Open one hint at a time.

Hint 1

Some inside terms have coefficients, like 2x. Multiply the outside number by the WHOLE term: 3 × 2x.

Hint 2

Two products per row: outside × first term, outside × second term. Keep the − sign if the inside subtracts.

Hint 3

Row 1: 3(2x + 5) → 3 × 2x = 6x. Now multiply 3 × 5 and join the products with +. Repeat for each row.

Check my answer
✅ Answers:
  • 3(2x + 5) → Expanded Form: 6x + 15
  • 7(a − 4) → Expanded Form: 7a − 28
  • 2(3m + 8) → Expanded Form: 6m + 16
  • 5(6 − y) → Expanded Form: 30 − 5y
The outside factor multiplies the whole term inside, coefficient included, and the sign inside stays the same. So 3(2x + 5) = 6x + 15, 7(a − 4) = 7a − 28, 2(3m + 8) = 6m + 16, and 5(6 − y) = 30 − 5y.

Turn & TalkSay it out loud with a partner

Each bundle is a ticket ($15) and a snack ($5). For 3 people, why does 3(15 + 5) give the same $60 as 3 × 15 + 3 × 5?

Try starting with:

  • Both give $60 because I distribute the 3 to ___ and ___.Las dos dan $60 porque distribuyo el 3 a ___ y ___.
  • So 3(15 + 5) equals ___ because ___.Entonces 3(15 + 5) es igual a ___ porque ___.
Use these wordsDistributive PropertyFactorExpandEquivalent